1. Technical Field
The present disclosure relates generally to the logging of subsurface formations surrounding a wellbore using a downhole logging tool, and particularly to making measurements with a modular logging tool while drilling, and using those measurements to infer one or more formation properties.
2. Background Art
Logging tools have long been used in wellbores to make, for example, formation evaluation measurements to infer properties of the formations surrounding the borehole and the fluids in the formations. Common logging tools include electromagnetic tools, nuclear tools, and nuclear magnetic resonance (NMR) tools, though various other tool types are also used.
Early logging tools were run into a wellbore on a wireline cable, after the wellbore had been drilled. Modern versions of such wireline tools are still used extensively. However, the need for information while drilling the borehole gave rise to measurement-while-drilling (MWD) tools and logging-while-drilling (LWD) tools. MWD tools typically provide drilling parameter information such as weight on the bit, torque, temperature, pressure, direction, and inclination. LWD tools typically provide formation evaluation measurements such as resistivity, porosity, and NMR distributions (e.g., T1 and T2). MWD and LWD tools often have components common to wireline tools (e.g., transmitting and receiving antennas), but MWD and LWD tools must be constructed to not only endure but to operate in the harsh environment of drilling.
Electromagnetic (EM) wave propagation in a medium is characterized by the magnetic permeability of the medium (μ) and the complex dielectric permittivity (∈*) given by,
                              ɛ          *                =                              ɛ            r                    -                      ⅈ            ⁢                          σ                              ϖɛ                0                                      ⁢                                                  ⁢            and                                              (        1        )                                μ        =                              μ            r                    ⁢                                    μ              0                        .                                              (        2        )            ∈r and μr are the permittivity and permeability of the medium relative to their corresponding values in free space (∈0=8.8 10−12, and μ0=1/(4π 10−7)), ω is the angular frequency, and σ is the conductivity. Those parameters affect the wave vector k, given by,
                    k        =                              ϖ            c                    ⁢                                                    μ                r                            ⁢                              ɛ                r                                                                        (        3        )            where c, the speed of light in vacuum, is given by,
                    c        =                              1                                                            μ                  0                                ⁢                                  ɛ                  0                                                              .                                    (        4        )            
Most rocks of interest are non-magnetic and therefore μr equals one. An EM measurement from a resistivity logging tool is related to k, which in turn is related to ∈r and σ. The real and imaginary parts of ∈* have different frequency dependencies. For example, the conductivity is typically constant until the frequency is above about 1 MHz, after which it increases slowly. The permittivity of rocks, on the other hand is very large (e.g., ˜10) at sub-Hz frequencies, and decreases as the frequency increases, but eventually flattens out at frequencies around a GHz. The frequency dependence of permittivity is 1/f for frequencies up to approximately 104 Hz, but between 104 and 108 Hz, it varies as 1/(fα), where α is approximately 0.3. Since the imaginary part of ∈* has an explicit 1/f dependence, the imaginary part dominates at low frequency and the real part dominates at high frequencies.
Most prior art low frequency resistivity tools have concentrated on the conductivity term of the complex permittivity and ignored the real part (which is known as the dielectric constant). As such, those tools only measure the amplitude of the received signal, which is sufficient to solve for the conductivity. However, if the phase of the received signal is also measured, one can additionally solve for the real and imaginary part of the complex permittivity. There is increasingly more interest in the dielectric constant since it contains information on the micro-geometry of the rock matrix.
Physics-based models explaining the frequency dependence of permittivity (and specifically the dielectric constant) attribute the variation with frequency to three effects, each of which operates in a particular frequency range. At high frequencies, where the permittivity is essentially frequency independent, the permittivity of the rock, which is a mixture of the solid matrix, water, and hydrocarbons, can be calculated using the “complex refractive index method” (CRIM), shown by Equation (5) below,√{square root over (∈*rock)}=(1−φ)√{square root over (∈matrix)}+Swaterφ√{square root over (∈*Water)}+(1−Swater)φ√{square root over (∈hydrocarbon)}.  (5)This is a simple volumetric average of the refractive index (that is, the square root of the permittivity) of the components. Any slight frequency dependence in this range is the result of the frequency dependence of the water permittivity.
The intermediate range, where the permittivity varies as the (−α) power of frequency, is attributed to the geometrical shape of the rock grains. The insulating grains, surrounded by conductive water, form local capacitors that respond to the applied electric field. The permittivity in this range has been described by several models, one of which, for a fully water-filled rock, is given by,
                    ϕ        =                              (                                                            ɛ                  rock                  *                                -                                  ɛ                  matrix                                                                              ɛ                  water                  *                                -                                  ɛ                  matrix                                                      )                    ⁢                                    (                                                ɛ                  water                  *                                                  ɛ                  rock                  *                                            )                        L                                              (        6        )            where L is the depolarizing factor describing the average grain shape. For example, L is ⅓ for spherical grains, and it deviates for more realistic, spheroidal grain shapes, though it remains between 0 and 1. This equation can be easily modified to include partial water saturation and the effect of hydrocarbons on the measured complex permittivity. As mentioned above, the intermediate frequency range starts at approximately 100 kHz, which is the operating frequency of most propagation and induction tools, so this expression is very applicable to the measurements from these tools and leads to a complex permittivity of water from which water salinity can be determined. The expression also provides a measure of grain shape that has further application.
At frequencies below 100 kHz, the permittivity has a 1/f dependence. This is attributed to the double layer effects caused by surface charges on the surfaces of the rock grains. The surfaces of the rock grains are charged either by the nature of the minerals at the surface, or, more importantly, by the varying amounts of clay mineral at the surface. These minerals have surface charges in contact with a cloud of oppositely charged counter-ions, forming an ionic double layer. The counter-ions respond to the applied electric field and cause a large permittivity. The permittivity in this frequency range is a clay indicator and can be used to estimate the clay concentration or cation-exchange capacity (CEC). Thus, any resistivity tool that measures the amplitude and phase of the received signal below 100 kHz can determine the conductivity and permittivity of the rock and can provide an estimate of the shale content. In addition to shale estimation, phase measurement may be used to determine a phase conductivity in addition to the traditionally measured amplitude conductivity. It has been shown that those two responses have different depths of investigations, and their combination provides a very good bed boundary indicator.